We report an experimental study of the magnetic field ~BB induced by a turbulent swirling flow of liquid sodium submitted to a transverse magnetic field ~BB0. We show that the induced field can behave nonlinearly as a function of the magnetic Reynolds number, Rm. At low Rm, the induced mean field along the axis of the flow, hBxi, and the one parallel to ~BB0, hByi, first behave like R2
m, whereas the third component, hBzi, is linear in Rm. The sign of hBxi is determined by the flow helicity. At higher Rm, ~BB
strongly depends on the local geometry of the mean flow: hBxi decreases to zero in the core of the swirling flow but remains finite outside. We compare the experimental results with the computed magnetic induction due to the mean flow alone.